8.1 Graphs of the sine and cosine functions (Page 3/13)

Page 3 / 13

A graph with four items. The x-axis ranges from -6pi to 6pi. The y-axis ranges from -4 to 4. The first item is the graph of sin(x), which has an amplitude of 1. The second is a graph of 2sin(x), which has amplitude of 2. The third is a graph of 3sin(x), which has an amplitude of 3. The fourth is a graph of 4 sin(x) with an amplitude of 4.

A General Note Label

Amplitude of sinusoidal functions

If we let $C = 0$ and $D = 0$ in the general form equations of the sine and cosine functions, we obtain the forms

y = A \sin (B x) and y = A \cos (B x)

The amplitude is $A,$ and the vertical height from the midline is $| A | .$ In addition, notice in the example that

| A | = amplitude = \frac{1}{2} | maximum - minimum |

Identifying the amplitude of a sine or cosine function

What is the amplitude of the sinusoidal function $f (x) = −4 \sin (x) ?$ Is the function stretched or compressed vertically?

Let’s begin by comparing the function to the simplified form $y = A \sin (B x) .$

In the given function, $A = −4,$ so the amplitude is $| A | = | −4 | = 4.$ The function is stretched.

Got questions? Get instant answers now!

Try IT Feature

What is the amplitude of the sinusoidal function $f (x) = \frac{1}{2} \sin (x) ?$ Is the function stretched or compressed vertically?

$\frac{1}{2}$ compressed

Got questions? Get instant answers now!

Analyzing graphs of variations of y = sin x And y = cos x

Now that we understand how $A$ and $B$ relate to the general form equation for the sine and cosine functions, we will explore the variables $C$ and $D .$ Recall the general form:

\begin{matrix} y = A \sin (B x - C) + D and y = A \cos (B x - C) + D \\ o r \\ y = A \sin (B (x - \frac{C}{B})) + D and y = A \cos (B (x - \frac{C}{B})) + D \end{matrix}

The value $\frac{C}{B}$ for a sinusoidal function is called the phase shift , or the horizontal displacement of the basic sine or cosine function . If $C > 0,$ the graph shifts to the right. If $C < 0,$ the graph shifts to the left. The greater the value of $| C |,$ the more the graph is shifted. [link] shows that the graph of $f (x) = \sin (x - π)$ shifts to the right by $π$ units, which is more than we see in the graph of $f (x) = \sin (x - \frac{π}{4}),$ which shifts to the right by $\frac{π}{4}$ units.

A graph with three items. The first item is a graph of sin(x). The second item is a graph of sin(x-pi/4), which is the same as sin(x) except shifted to the right by pi/4. The third item is a graph of sin(x-pi), which is the same as sin(x) except shifted to the right by pi.

While $C$ relates to the horizontal shift, $D$ indicates the vertical shift from the midline in the general formula for a sinusoidal function. See [link] . The function $y = \cos (x) + D$ has its midline at $y = D .$

A graph of y=Asin(x)+D. Graph shows the midline of the function at y=D.

Any value of $D$ other than zero shifts the graph up or down. [link] compares $f (x) = \sin x$ with $f (x) = \sin x + 2,$ which is shifted 2 units up on a graph.

A graph with two items. The first item is a graph of sin(x). The second item is a graph of sin(x)+2, which is the same as sin(x) except shifted up by 2.

A General Note label

Variations of sine and cosine functions

Given an equation in the form $f (x) = A \sin (B x - C) + D$ or $f (x) = A \cos (B x - C) + D,$ $\frac{C}{B}$ is the phase shift and $D$ is the vertical shift .

Identifying the phase shift of a function

Determine the direction and magnitude of the phase shift for $f (x) = \sin (x + \frac{π}{6}) - 2.$

Let’s begin by comparing the equation to the general form $y = A \sin (B x - C) + D .$

In the given equation, notice that $B = 1$ and $C = - \frac{π}{6} .$ So the phase shift is

\begin{array}{r} \frac{C}{B} = - \frac{\frac{π}{6}}{1} \\ = - \frac{π}{6} \end{array}

or $\frac{π}{6}$ units to the left.

Got questions? Get instant answers now!

Try IT Feature

Determine the direction and magnitude of the phase shift for $f (x) = 3 \cos (x - \frac{π}{2}) .$

$\frac{π}{2};$ right

Got questions? Get instant answers now!

Identifying the vertical shift of a function

Determine the direction and magnitude of the vertical shift for $f (x) = \cos (x) - 3.$

Let’s begin by comparing the equation to the general form $y = A \cos (B x - C) + D .$

In the given equation, $D = −3$ so the shift is 3 units downward.

Got questions? Get instant answers now!

Try IT Feature

Determine the direction and magnitude of the vertical shift for $f (x) = 3 \sin (x) + 2.$

2 units up

Got questions? Get instant answers now!

How to Feature

Given a sinusoidal function in the form $f (x) = A \sin (B x - C) + D,$ identify the midline, amplitude, period, and phase shift.

Determine the amplitude as $| A | .$
Determine the period as $P = \frac{2 π}{| B |} .$
Determine the phase shift as $\frac{C}{B} .$
Determine the midline as $y = D .$

Questions & Answers

differentiate between demand and supply giving examples

Lambiv Reply

differentiated between demand and supply using examples

Lambiv

what is labour ?

Lambiv

how will I do?

Venny Reply

how is the graph works?I don't fully understand

Rezat Reply

information

Eliyee

devaluation

Eliyee

WARKISA

hi guys good evening to all

Lambiv

multiple choice question

Aster Reply

appreciation

Eliyee

explain perfect market

Lindiwe Reply

In economics, a perfect market refers to a theoretical construct where all participants have perfect information, goods are homogenous, there are no barriers to entry or exit, and prices are determined solely by supply and demand. It's an idealized model used for analysis,

Ezea

What is ceteris paribus?

Shukri Reply

other things being equal

AI-Robot

When MP₁ becomes negative, TP start to decline. Extuples Suppose that the short-run production function of certain cut-flower firm is given by: Q=4KL-0.6K2 - 0.112 • Where is quantity of cut flower produced, I is labour input and K is fixed capital input (K-5). Determine the average product of lab

Kelo

Extuples Suppose that the short-run production function of certain cut-flower firm is given by: Q=4KL-0.6K2 - 0.112 • Where is quantity of cut flower produced, I is labour input and K is fixed capital input (K-5). Determine the average product of labour (APL) and marginal product of labour (MPL)

Kelo

yes,thank you

Shukri

Can I ask you other question?

Shukri

what is monopoly mean?

Habtamu Reply

What is different between quantity demand and demand?

Shukri Reply

Quantity demanded refers to the specific amount of a good or service that consumers are willing and able to purchase at a give price and within a specific time period. Demand, on the other hand, is a broader concept that encompasses the entire relationship between price and quantity demanded

Ezea

Shukri

how do you save a country economic situation when it's falling apart

Lilia Reply

what is the difference between economic growth and development

Fiker Reply

Economic growth as an increase in the production and consumption of goods and services within an economy.but Economic development as a broader concept that encompasses not only economic growth but also social & human well being.

Shukri

production function means

Jabir

What do you think is more important to focus on when considering inequality ?

Abdisa Reply

any question about economics?

Awais Reply

sir...I just want to ask one question... Define the term contract curve? if you are free please help me to find this answer 🙏

Asui

it is a curve that we get after connecting the pareto optimal combinations of two consumers after their mutually beneficial trade offs

Awais

thank you so much 👍 sir

Asui

In economics, the contract curve refers to the set of points in an Edgeworth box diagram where both parties involved in a trade cannot be made better off without making one of them worse off. It represents the Pareto efficient allocations of goods between two individuals or entities, where neither p

Cornelius

Suppose a consumer consuming two commodities X and Y has The following utility function u=X0.4 Y0.6. If the price of the X and Y are 2 and 3 respectively and income Constraint is birr 50. A,Calculate quantities of x and y which maximize utility. B,Calculate value of Lagrange multiplier. C,Calculate quantities of X and Y consumed with a given price. D,alculate optimum level of output .

Feyisa Reply

Answer

Feyisa

Jabir

the market for lemon has 10 potential consumers, each having an individual demand curve p=101-10Qi, where p is price in dollar's per cup and Qi is the number of cups demanded per week by the i th consumer.Find the market demand curve using algebra. Draw an individual demand curve and the market dema

Gsbwnw Reply

suppose the production function is given by ( L, K)=L¼K¾.assuming capital is fixed find APL and MPL. consider the following short run production function:Q=6L²-0.4L³ a) find the value of L that maximizes output b)find the value of L that maximizes marginal product

Abdureman

types of unemployment

Yomi Reply

What is the difference between perfect competition and monopolistic competition?

Mohammed

Got questions? Join the online conversation and get instant answers!

Jobilize.com Reply

<< Chapter < Page Page > Chapter >>

Practice Key Terms 5

Read also:

Get Jobilize Job Search Mobile App in your pocket Now!

100% Free Mobile Applications
Receive real-time job alerts and never miss the right job again

Source: OpenStax, Algebra and trigonometry. OpenStax CNX. Nov 14, 2016 Download for free at https://legacy.cnx.org/content/col11758/1.6

Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Algebra and trigonometry' conversation and receive update notifications?

Ask

	6 Graphing variations of y = sin x And y = cos x By OpenStax Read Online Course
	4 Determining amplitude By OpenStax Read Online Course
	9 Arts Society: Theater 9 Exam By Jonathan Long Start Exam
	Anthropology Aesthetics Culture By Richley Crapo Start Assignment
©flickr: Bertram	Chemistry Ch 1 2 Test 1 By Madison Christian Start Quiz
	Anthropology Environment, Adaption Subsistence By Richley Crapo Start Assignment
	Introductory statistics By OpenStax Read Online Course
	Are you a 5sos fan? By Rylee Minllic Start Quiz
	Social Organization Kinship By Richley Crapo Start Assignment
	46 Biology 46 Ecosystems MCQ By OpenStax Start Quiz
	18 AP 18 Cardiovascular System Blood MCQ By OpenStax Start Quiz
	Precalculus By OpenStax Read Online Course